Copied to
clipboard

G = C10×C22⋊Q8order 320 = 26·5

Direct product of C10 and C22⋊Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C10×C22⋊Q8, C233(C5×Q8), C4.63(D4×C10), (C22×C10)⋊6Q8, C221(Q8×C10), (C2×C20).524D4, C20.470(C2×D4), (C22×Q8)⋊3C10, C24.31(C2×C10), (C23×C20).25C2, (C23×C4).10C10, (Q8×C10)⋊48C22, C22.60(D4×C10), C10.57(C22×Q8), (C2×C10).343C24, (C2×C20).656C23, C10.182(C22×D4), (C23×C10).91C22, C23.70(C22×C10), C22.17(C23×C10), (C22×C20).444C22, (C22×C10).258C23, C2.6(D4×C2×C10), C2.3(Q8×C2×C10), (C10×C4⋊C4)⋊42C2, (C2×C4⋊C4)⋊15C10, (Q8×C2×C10)⋊15C2, (C2×C10)⋊5(C2×Q8), C4⋊C410(C2×C10), (C2×Q8)⋊8(C2×C10), C2.6(C10×C4○D4), (C5×C4⋊C4)⋊66C22, (C2×C4).135(C5×D4), C10.225(C2×C4○D4), (C2×C10).682(C2×D4), C22.30(C5×C4○D4), (C10×C22⋊C4).31C2, (C2×C22⋊C4).11C10, C22⋊C4.10(C2×C10), (C2×C4).12(C22×C10), (C22×C4).53(C2×C10), (C2×C10).230(C4○D4), (C5×C22⋊C4).144C22, SmallGroup(320,1525)

Series: Derived Chief Lower central Upper central

C1C22 — C10×C22⋊Q8
C1C2C22C2×C10C2×C20Q8×C10C5×C22⋊Q8 — C10×C22⋊Q8
C1C22 — C10×C22⋊Q8
C1C22×C10 — C10×C22⋊Q8

Generators and relations for C10×C22⋊Q8
 G = < a,b,c,d,e | a10=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 450 in 322 conjugacy classes, 194 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C5, C2×C4 [×16], C2×C4 [×18], Q8 [×8], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C24, C20 [×4], C20 [×10], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×8], C23×C4, C22×Q8, C2×C20 [×16], C2×C20 [×18], C5×Q8 [×8], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22⋊Q8, C5×C22⋊C4 [×8], C5×C4⋊C4 [×12], C22×C20 [×2], C22×C20 [×8], C22×C20 [×4], Q8×C10 [×4], Q8×C10 [×4], C23×C10, C10×C22⋊C4 [×2], C10×C4⋊C4, C10×C4⋊C4 [×2], C5×C22⋊Q8 [×8], C23×C20, Q8×C2×C10, C10×C22⋊Q8
Quotients: C1, C2 [×15], C22 [×35], C5, D4 [×4], Q8 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C2×C10 [×35], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C5×D4 [×4], C5×Q8 [×4], C22×C10 [×15], C2×C22⋊Q8, D4×C10 [×6], Q8×C10 [×6], C5×C4○D4 [×2], C23×C10, C5×C22⋊Q8 [×4], D4×C2×C10, Q8×C2×C10, C10×C4○D4, C10×C22⋊Q8

Smallest permutation representation of C10×C22⋊Q8
On 160 points
Generators in S160
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(31 160)(32 151)(33 152)(34 153)(35 154)(36 155)(37 156)(38 157)(39 158)(40 159)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 121)(118 122)(119 123)(120 124)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 61)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(31 160)(32 151)(33 152)(34 153)(35 154)(36 155)(37 156)(38 157)(39 158)(40 159)(41 59)(42 60)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(49 57)(50 58)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(91 107)(92 108)(93 109)(94 110)(95 101)(96 102)(97 103)(98 104)(99 105)(100 106)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 121)(118 122)(119 123)(120 124)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)
(1 79 56 107)(2 80 57 108)(3 71 58 109)(4 72 59 110)(5 73 60 101)(6 74 51 102)(7 75 52 103)(8 76 53 104)(9 77 54 105)(10 78 55 106)(11 146 34 118)(12 147 35 119)(13 148 36 120)(14 149 37 111)(15 150 38 112)(16 141 39 113)(17 142 40 114)(18 143 31 115)(19 144 32 116)(20 145 33 117)(21 140 153 122)(22 131 154 123)(23 132 155 124)(24 133 156 125)(25 134 157 126)(26 135 158 127)(27 136 159 128)(28 137 160 129)(29 138 151 130)(30 139 152 121)(41 94 65 86)(42 95 66 87)(43 96 67 88)(44 97 68 89)(45 98 69 90)(46 99 70 81)(47 100 61 82)(48 91 62 83)(49 92 63 84)(50 93 64 85)
(1 119 56 147)(2 120 57 148)(3 111 58 149)(4 112 59 150)(5 113 60 141)(6 114 51 142)(7 115 52 143)(8 116 53 144)(9 117 54 145)(10 118 55 146)(11 78 34 106)(12 79 35 107)(13 80 36 108)(14 71 37 109)(15 72 38 110)(16 73 39 101)(17 74 40 102)(18 75 31 103)(19 76 32 104)(20 77 33 105)(21 82 153 100)(22 83 154 91)(23 84 155 92)(24 85 156 93)(25 86 157 94)(26 87 158 95)(27 88 159 96)(28 89 160 97)(29 90 151 98)(30 81 152 99)(41 134 65 126)(42 135 66 127)(43 136 67 128)(44 137 68 129)(45 138 69 130)(46 139 70 121)(47 140 61 122)(48 131 62 123)(49 132 63 124)(50 133 64 125)

G:=sub<Sym(160)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,61)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(41,59)(42,60)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,107)(92,108)(93,109)(94,110)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146), (1,79,56,107)(2,80,57,108)(3,71,58,109)(4,72,59,110)(5,73,60,101)(6,74,51,102)(7,75,52,103)(8,76,53,104)(9,77,54,105)(10,78,55,106)(11,146,34,118)(12,147,35,119)(13,148,36,120)(14,149,37,111)(15,150,38,112)(16,141,39,113)(17,142,40,114)(18,143,31,115)(19,144,32,116)(20,145,33,117)(21,140,153,122)(22,131,154,123)(23,132,155,124)(24,133,156,125)(25,134,157,126)(26,135,158,127)(27,136,159,128)(28,137,160,129)(29,138,151,130)(30,139,152,121)(41,94,65,86)(42,95,66,87)(43,96,67,88)(44,97,68,89)(45,98,69,90)(46,99,70,81)(47,100,61,82)(48,91,62,83)(49,92,63,84)(50,93,64,85), (1,119,56,147)(2,120,57,148)(3,111,58,149)(4,112,59,150)(5,113,60,141)(6,114,51,142)(7,115,52,143)(8,116,53,144)(9,117,54,145)(10,118,55,146)(11,78,34,106)(12,79,35,107)(13,80,36,108)(14,71,37,109)(15,72,38,110)(16,73,39,101)(17,74,40,102)(18,75,31,103)(19,76,32,104)(20,77,33,105)(21,82,153,100)(22,83,154,91)(23,84,155,92)(24,85,156,93)(25,86,157,94)(26,87,158,95)(27,88,159,96)(28,89,160,97)(29,90,151,98)(30,81,152,99)(41,134,65,126)(42,135,66,127)(43,136,67,128)(44,137,68,129)(45,138,69,130)(46,139,70,121)(47,140,61,122)(48,131,62,123)(49,132,63,124)(50,133,64,125)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,61)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(41,59)(42,60)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,107)(92,108)(93,109)(94,110)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146), (1,79,56,107)(2,80,57,108)(3,71,58,109)(4,72,59,110)(5,73,60,101)(6,74,51,102)(7,75,52,103)(8,76,53,104)(9,77,54,105)(10,78,55,106)(11,146,34,118)(12,147,35,119)(13,148,36,120)(14,149,37,111)(15,150,38,112)(16,141,39,113)(17,142,40,114)(18,143,31,115)(19,144,32,116)(20,145,33,117)(21,140,153,122)(22,131,154,123)(23,132,155,124)(24,133,156,125)(25,134,157,126)(26,135,158,127)(27,136,159,128)(28,137,160,129)(29,138,151,130)(30,139,152,121)(41,94,65,86)(42,95,66,87)(43,96,67,88)(44,97,68,89)(45,98,69,90)(46,99,70,81)(47,100,61,82)(48,91,62,83)(49,92,63,84)(50,93,64,85), (1,119,56,147)(2,120,57,148)(3,111,58,149)(4,112,59,150)(5,113,60,141)(6,114,51,142)(7,115,52,143)(8,116,53,144)(9,117,54,145)(10,118,55,146)(11,78,34,106)(12,79,35,107)(13,80,36,108)(14,71,37,109)(15,72,38,110)(16,73,39,101)(17,74,40,102)(18,75,31,103)(19,76,32,104)(20,77,33,105)(21,82,153,100)(22,83,154,91)(23,84,155,92)(24,85,156,93)(25,86,157,94)(26,87,158,95)(27,88,159,96)(28,89,160,97)(29,90,151,98)(30,81,152,99)(41,134,65,126)(42,135,66,127)(43,136,67,128)(44,137,68,129)(45,138,69,130)(46,139,70,121)(47,140,61,122)(48,131,62,123)(49,132,63,124)(50,133,64,125) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(31,160),(32,151),(33,152),(34,153),(35,154),(36,155),(37,156),(38,157),(39,158),(40,159),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,121),(118,122),(119,123),(120,124),(131,147),(132,148),(133,149),(134,150),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,61),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(31,160),(32,151),(33,152),(34,153),(35,154),(36,155),(37,156),(38,157),(39,158),(40,159),(41,59),(42,60),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(49,57),(50,58),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,81),(78,82),(79,83),(80,84),(91,107),(92,108),(93,109),(94,110),(95,101),(96,102),(97,103),(98,104),(99,105),(100,106),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,121),(118,122),(119,123),(120,124),(131,147),(132,148),(133,149),(134,150),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146)], [(1,79,56,107),(2,80,57,108),(3,71,58,109),(4,72,59,110),(5,73,60,101),(6,74,51,102),(7,75,52,103),(8,76,53,104),(9,77,54,105),(10,78,55,106),(11,146,34,118),(12,147,35,119),(13,148,36,120),(14,149,37,111),(15,150,38,112),(16,141,39,113),(17,142,40,114),(18,143,31,115),(19,144,32,116),(20,145,33,117),(21,140,153,122),(22,131,154,123),(23,132,155,124),(24,133,156,125),(25,134,157,126),(26,135,158,127),(27,136,159,128),(28,137,160,129),(29,138,151,130),(30,139,152,121),(41,94,65,86),(42,95,66,87),(43,96,67,88),(44,97,68,89),(45,98,69,90),(46,99,70,81),(47,100,61,82),(48,91,62,83),(49,92,63,84),(50,93,64,85)], [(1,119,56,147),(2,120,57,148),(3,111,58,149),(4,112,59,150),(5,113,60,141),(6,114,51,142),(7,115,52,143),(8,116,53,144),(9,117,54,145),(10,118,55,146),(11,78,34,106),(12,79,35,107),(13,80,36,108),(14,71,37,109),(15,72,38,110),(16,73,39,101),(17,74,40,102),(18,75,31,103),(19,76,32,104),(20,77,33,105),(21,82,153,100),(22,83,154,91),(23,84,155,92),(24,85,156,93),(25,86,157,94),(26,87,158,95),(27,88,159,96),(28,89,160,97),(29,90,151,98),(30,81,152,99),(41,134,65,126),(42,135,66,127),(43,136,67,128),(44,137,68,129),(45,138,69,130),(46,139,70,121),(47,140,61,122),(48,131,62,123),(49,132,63,124),(50,133,64,125)])

140 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P5A5B5C5D10A···10AB10AC···10AR20A···20AF20AG···20BL
order12···222224···44···4555510···1010···1020···2020···20
size11···122222···24···411111···12···22···24···4

140 irreducible representations

dim111111111111222222
type+++++++-
imageC1C2C2C2C2C2C5C10C10C10C10C10D4Q8C4○D4C5×D4C5×Q8C5×C4○D4
kernelC10×C22⋊Q8C10×C22⋊C4C10×C4⋊C4C5×C22⋊Q8C23×C20Q8×C2×C10C2×C22⋊Q8C2×C22⋊C4C2×C4⋊C4C22⋊Q8C23×C4C22×Q8C2×C20C22×C10C2×C10C2×C4C23C22
# reps12381148123244444161616

Matrix representation of C10×C22⋊Q8 in GL5(𝔽41)

400000
018000
001800
000250
000025
,
400000
01000
004000
00010
00001
,
10000
040000
004000
00010
00001
,
10000
01000
00100
0004039
00011
,
10000
004000
040000
0003414
000147

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,25,0,0,0,0,0,25],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,1,0,0,0,39,1],[1,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,34,14,0,0,0,14,7] >;

C10×C22⋊Q8 in GAP, Magma, Sage, TeX

C_{10}\times C_2^2\rtimes Q_8
% in TeX

G:=Group("C10xC2^2:Q8");
// GroupNames label

G:=SmallGroup(320,1525);
// by ID

G=gap.SmallGroup(320,1525);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,568,3446]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽